Spectral Fukaya Category Overview

• Spectral Fukaya category is a framework that refines traditional Fukaya theory by incorporating additional homotopical, monodromic, or eigenvalue data.
• It encompasses enrichment over spectra, eigenvalue decomposition of quantum cohomology, and deformation-theoretic enhancements using B-fields and divisor data.
• Applications include advances in mirror symmetry, Weinstein localization, and computational approaches via sheaf-theoretic methods and homotopy colimits.

Spectral Fukaya category denotes a family of related refinements of Fukaya-theoretic constructions rather than a single universally fixed object. In the cited literature, the phrase is used for at least four closely connected patterns: a Fukaya category enriched over bordism groups or spectra; a decomposition of a Fukaya category into summands indexed by the spectrum of quantum multiplication or by critical values of a Landau–Ginzburg potential; a sheaf of stable $\infty$-categories on a Weinstein manifold; and a deformation-theoretic package in which $A_\infty$-operations vary under $B$-fields, divisor data, or symplectic-cohomological Maurer–Cartan elements. This suggests that “spectral Fukaya category” functions as an umbrella term for stable-homotopical, monodromic, or eigenvalue-sensitive enhancements of Fukaya categories (Porcelli et al., 25 Sep 2025, Castronovo, 2018, Nadler, 2011, Rabah, 3 Jul 2026).

1. Terminological scope

The cited literature uses “spectral” in several technically distinct senses. Some papers mean an actual enrichment over spectra or bordism theories. Others mean a spectral decomposition by eigenvalues of $c_1\star$ or of a mirror superpotential. Others use “spectral” for categorical sheaf-theoretic localization or for stability data encoded by spectral networks. The common feature is that the Fukaya category is no longer treated as only an abstract chain-level $A_\infty$-category over a field; it is organized by additional homotopical, monodromic, or eigenvalue data.

Usage	Core datum	Representative source
Spectrum or bordism enrichment	Thom spectrum $R_\Psi$, morphisms $\Omega_*^{E_\Psi}(M^{LK})$	(Porcelli et al., 25 Sep 2025)
Eigenvalue decomposition	Summands $\mathcal F_\lambda(X)$, $QH^*(X)=\bigoplus_\lambda QH_\lambda(X)$	(Castronovo, 2018, Smith, 2018)
Sheaf of stable categories	$\mathcal F_\Lambda$ on the conic topology, $A_\infty$0	(Nadler, 2011)
Deformation-theoretic refinement	$A_\infty$1-field twisting, relative divisor variables $A_\infty$2, Maurer–Cartan deformation	(Azam et al., 2023, Rabah, 3 Jul 2026)

A recurrent misconception is that “spectral Fukaya category” always means enrichment over spectra in the strict homotopy-theoretic sense. The literature shows a broader usage. In toric and Grassmannian contexts, “spectral” can instead mean decomposition into generalized eigensummands of quantum cohomology; in Weinstein localization, it can mean a sheaf of stable categories; and in exact bordism-theoretic work it can mean an honest spectrum-level or Thom-spectrum-valued refinement (Castronovo, 2018, Nadler, 2011, Porcelli et al., 25 Sep 2025).

2. Spectrum-enriched and bordism-theoretic constructions

The most literal realization of the term appears in work on exact Liouville domains equipped with tangential data $A_\infty$3. There one constructs a spectral Fukaya category $A_\infty$4 whenever $A_\infty$5 lifts to $A_\infty$6, with objects $A_\infty$7 consisting of closed exact Lagrangians endowed with compatible $A_\infty$8-orientations. Morphisms are not Floer cochain complexes over a ring; they are bordism groups of $A_\infty$9-oriented flow modules over Floer flow categories,

$B$0

where $B$1 is a Thom $B$2-monoid and $B$3 is the associated commutative ring spectrum with bordism theory $B$4 (Porcelli et al., 25 Sep 2025).

In this model, tangential data are fundamental. The homotopy fibre of $B$5 determines a Thom $B$6-monoid, and Bott periodicity and index theory produce the ring spectrum $B$7. The resulting category is “spectral” in a strong sense: its enrichment is controlled by a Thom spectrum rather than by ordinary coefficients. Classical examples occur when $B$8, giving the sphere spectrum and framed bordism, or when $B$9, giving $c_1\star$0 and complex cobordism (Porcelli et al., 25 Sep 2025).

This framework also incorporates rank-one spectral local systems. A rank-one spectral local system is a map

$c_1\star$1

the spectral analogue of a rank-one local system of modules. Twisting by $c_1\star$2 modifies the flow categories and yields a local-system-enriched spectral Fukaya category. The associated open-closed class satisfies the explicit formula

$c_1\star$3

where $c_1\star$4 is a multiplicatively two-torsion unit arising from the action of the stable Hopf map $c_1\star$5. In contrast to classical exact Fukaya categories over $c_1\star$6, where the open-closed image of a brane is independent of the rank-one local system, the spectral local system contributes a nontrivial $c_1\star$7-primary correction (Porcelli et al., 25 Sep 2025).

This construction supplies a precise bordism-theoretic enhancement of exact Fukaya theory. It is also a template for how stable homotopy data can enter Fukaya categories without passing through a purely algebraic dg model first.

3. Spectral decomposition by quantum cohomology and mirror superpotentials

A second major meaning of “spectral Fukaya category” is decomposition into summands indexed by the spectrum of quantum multiplication. In the monotone setting one decomposes

$c_1\star$8

where $c_1\star$9 is the generalized eigenspace for $A_\infty$0 with eigenvalue $A_\infty$1. Correspondingly, one defines Fukaya summands $A_\infty$2 whose objects satisfy $A_\infty$3, so that

$A_\infty$4

In this usage, the spectral Fukaya category is the collection of these eigensummands and their mirror identification with fibres $A_\infty$5 of a Landau–Ginzburg superpotential $A_\infty$6 (Castronovo, 2018).

The Grassmannian case makes this formulation explicit. For $A_\infty$7, the eigenvalues of $A_\infty$8 are

$A_\infty$9

where $R_\Psi$0 runs over size-$R_\Psi$1 subsets of roots of $R_\Psi$2. The paper identifies the monotone Gelfand–Cetlin torus $R_\Psi$3, computes its Maslov $R_\Psi$4 disk potential, and shows that its holonomy local systems provide nonzero objects in the corresponding spectral summands. For $R_\Psi$5 with $R_\Psi$6 prime, all eigenvalues have multiplicity one, which yields a complete set of generators for all spectral summands and a fibrewise homological mirror symmetry statement

$R_\Psi$7

for every eigenvalue $R_\Psi$8 (Castronovo, 2018).

In compact toric varieties, the same pattern is organized through the decomposition

$R_\Psi$9

with each summand $\Omega_*^{E_\Psi}(M^{LK})$0 associated to a toric fibre $\Omega_*^{E_\Psi}(M^{LK})$1 equipped with a rank $\Omega_*^{E_\Psi}(M^{LK})$2 local system. The closed-open string map factors through the Kodaira–Spencer isomorphism on each summand, and, assuming an appropriate version of Abouzaid’s criterion, $\Omega_*^{E_\Psi}(M^{LK})$3 split-generates the corresponding summand of the Fukaya category (Smith, 2018). Here “spectral” refers not to stable homotopy types but to the spectrum of the quantum cohomology algebra and to the mirror critical-point decomposition of the Jacobian ring.

This eigenvalue-based usage is now standard in monotone mirror symmetry. It aligns Fukaya-theoretic decomposition with the decomposition of the mirror Landau–Ginzburg model into critical-value fibres, thereby turning the spectral data of $\Omega_*^{E_\Psi}(M^{LK})$4 into a categorical direct-sum structure.

4. Twists, deformations, and open–closed structures

A third cluster of meanings concerns controlled deformations of Fukaya categories. One source is $\Omega_*^{E_\Psi}(M^{LK})$5-field twisting. For a symplectic manifold $\Omega_*^{E_\Psi}(M^{LK})$6 with closed $\Omega_*^{E_\Psi}(M^{LK})$7-form $\Omega_*^{E_\Psi}(M^{LK})$8, the relevant complexified symplectic form is

$\Omega_*^{E_\Psi}(M^{LK})$9

Objects are Lagrangian branes $\mathcal F_\lambda(X)$0 such that

$\mathcal F_\lambda(X)$1

Morphisms are Floer complexes with coefficients in fibrewise homomorphisms of the line bundles, and the $\mathcal F_\lambda(X)$2-operations are weighted by

$\mathcal F_\lambda(X)$3

together with parallel transport along the boundary segments. The weight depends only on the homotopy class of the disk, and under a Lagrangian isotopy $\mathcal F_\lambda(X)$4 it changes by

$\mathcal F_\lambda(X)$5

Exact isotopies give trivial factors, while non-exact isotopies produce controlled twists by classes in $\mathcal F_\lambda(X)$6 (Azam et al., 2023).

This $\mathcal F_\lambda(X)$7-field formalism is not yet a spectrum-enriched category, but it supplies precisely the homotopy-invariant twisting and isotopy-control that a stable or parameterized refinement would require. The paper itself presents these ingredients as what one needs to think about “spectral Fukaya categories”: stable homotopy refinements, spectral enrichments, and twisted or parameterized variants (Azam et al., 2023).

A second deformation-theoretic source is the relative Fukaya category. For a closed monotone symplectic manifold $\mathcal F_\lambda(X)$8 with simple normal-crossings divisor $\mathcal F_\lambda(X)$9, one defines

$QH^*(X)=\bigoplus_\lambda QH_\lambda(X)$0

and constructs a relative Fukaya category $QH^*(X)=\bigoplus_\lambda QH_\lambda(X)$1 whose objects are the same compact exact branes as those of $QH^*(X)=\bigoplus_\lambda QH_\lambda(X)$2, $QH^*(X)=\bigoplus_\lambda QH_\lambda(X)$3, but whose $QH^*(X)=\bigoplus_\lambda QH_\lambda(X)$4-operations record intersection multiplicities with $QH^*(X)=\bigoplus_\lambda QH_\lambda(X)$5 through monomials $QH^*(X)=\bigoplus_\lambda QH_\lambda(X)$6. The associated deformation on the wrapped side is governed by a Maurer–Cartan element

$QH^*(X)=\bigoplus_\lambda QH_\lambda(X)$7

in an $QH^*(X)=\bigoplus_\lambda QH_\lambda(X)$8-algebra built from symplectic cohomology, and the closed–open $QH^*(X)=\bigoplus_\lambda QH_\lambda(X)$9-morphism transfers $\mathcal F_\Lambda$0 to Hochschild cochains of the wrapped category. The main theorem identifies the resulting deformed wrapped category on compact objects with the relative Fukaya category, proving Conjecture $\mathcal F_\Lambda$1 of SBEAS24.

Open–closed maps supply the algebraic backbone for these deformation pictures. In the wrapped exact setting, one has a map from Hochschild homology to symplectic cohomology, and if the identity $\mathcal F_\Lambda$2 lies in its image for a full subcategory, then that subcategory split-generates the wrapped Fukaya category (Abouzaid, 2010). This gives a precise mechanism by which closed-string “spectral” data control open-string generation.

Taken together, these constructions show that spectral Fukaya theory can also mean a deformation package: $\mathcal F_\Lambda$3-field phases, divisor variables, Maurer–Cartan elements in symplectic cohomology, and closed–open transfer to Hochschild cochains.

5. Local-to-global, sheaf-theoretic, and homotopy-colimit formulations

A fourth meaning of “spectral Fukaya category” is local-to-global organization by stable or dg-categorical descent. In Nadler’s categorical Morse-theoretic formulation for Weinstein manifolds, the perfect Fukaya category is not treated as a single indivisible object. Instead, one associates local Fukaya categories to Weinstein cells and glues them by recollement. For a marked Weinstein manifold $\mathcal F_\Lambda$4, one obtains a sheaf $\mathcal F_\Lambda$5 of stable $\mathcal F_\Lambda$6-linear $\mathcal F_\Lambda$7-categories on the conic topology of $\mathcal F_\Lambda$8, supported on the characteristic cone $\mathcal F_\Lambda$9, with

$A_\infty$00

Locally above each unstable cell $A_\infty$01, the sheaf restricts to a category equivalent to $A_\infty$02, where $A_\infty$03 is the Weinstein cell obtained by Hamiltonian reduction (Nadler, 2011).

This is “spectral” in the sense of higher-categorical localization. The Fukaya category becomes the global sections of a sheaf of stable $A_\infty$04-categories, and cell-wise recollement behaves as a categorified Morse decomposition. The paper also suggests a dual cosheaf picture for partially wrapped categories, further reinforcing the local-to-global interpretation (Nadler, 2011).

A more computational dg-categorical realization appears in work on homotopy colimits. For semifree dg categories, one constructs an explicit cylinder object and an explicit formula for homotopy colimits in $A_\infty$05. Combined with the sectorial descent theorem of Ganatra–Pardon–Shende, this gives practical formulas computing wrapped Fukaya categories of Weinstein manifolds from sectorial coverings. In particular, wrapped Fukaya categories of cotangent bundles and plumbing spaces can be computed by dg homotopy colimits, and for lens spaces the endomorphism algebra of the cotangent fibre detects the homotopy type (Karabas et al., 2021).

The sheaf-theoretic realization of Fukaya categories at infinity provides another nearby development. For Legendrian knots in the contact boundary of a cotangent bundle, the relevant Fukaya category is equivalent to a dg category of constructible sheaves with singular support controlled by the front projection. In positive braid situations, weight filtrations on pushforwards from moduli spaces of such objects produce spectral sequences whose $A_\infty$06-page is colored triply graded Khovanov–Rozansky homology (Shende et al., 2014). This does not use “spectral Fukaya category” in the same way as spectrum enrichment, but it shows how categorical localization can produce genuinely spectral invariants.

These local-to-global pictures replace a single chain complex by stable gluing data, recollement, and homotopy colimits. In that sense they are among the most structurally “spectral” approaches in the current literature.

6. Monodromy, stability conditions, and adjacent programs

Lefschetz fibrations provide another source of spectral structure. For the directed Fukaya algebra $A_\infty$07 of vanishing thimbles in an exact symplectic Lefschetz fibration, the monodromy around infinity $A_\infty$08 yields a canonical map

$A_\infty$09

so classes in fixed-point Floer cohomology produce natural transformations from the Serre functor to the identity. In the anticanonical pencil case, one obtains a distinguished pair $A_\infty$10 of bimodule maps $A_\infty$11, and these are organized by Seidel into the framework of noncommutative divisors and pencils (Seidel, 2014). Here “spectral” refers to monodromy data, Serre functor sections, and the family of categorical structures parametrized by the pencil.

A different adjacent program appears in Fukaya categories with coefficients. Given a surface, a triangulated dg-category $A_\infty$12, and a holomorphic family of Bridgeland stability conditions on $A_\infty$13, one defines spectral networks as objects in a Fukaya category of the surface with coefficients in $A_\infty$14. In the constant-family case, a spectral network of phase $A_\infty$15 is a graph-supported object whose fibre labels are semistable and whose total phase is constant; its central charge is

$A_\infty$16

The paper conjectures that semistable objects of the resulting Fukaya category with coefficients are precisely those admitting spectral network representatives, proves a uniqueness theorem up to $A_\infty$17-equivalence of the fibre data, and verifies the conjecture for a disk with six marked boundary points and coefficient category $A_\infty$18 (Haiden et al., 2021).

These programs clarify why the term remains nonuniform. “Spectral Fukaya category” may denote a bordism-enriched category, an eigenvalue decomposition, a monodromy-controlled Lefschetz-theoretic structure, or a stability-theoretic surface category with spectral-network representatives. The cited literature therefore supports a cautious encyclopedia definition: the phrase refers to a family of constructions in which Fukaya categories are refined by spectra, by spectral decompositions, by sheaf-theoretic localization, or by deformation and stability data, with mirror symmetry providing the principal unifying context (Porcelli et al., 25 Sep 2025, Castronovo, 2018, Seidel, 2014, Haiden et al., 2021).